“Exploratory Experimentation and Computation” published in AMS Notices

An article entitled “Exploratory Experimentation and Computation,” authored by the present bloggers, has appeared in the November 2011 issue of the Notices of the American Mathematical Society. The full PDF of the article is available Here. The article has been highlighted in a number of press reports, including: LBNL News, Science Daily, Eurekalert, Physorg, Newswise, and Others.

A common misperception is that mathematicians’ work consists entirely of calculations. If that were true, computers would have replaced mathematicians long ago. What mathematicians actually do is to discover and investigate patterns—patterns that arise in numbers, in abstract shapes, in transformations between different mathematical objects, and so on. Studying such patterns requires subtle and sophisticated tools, and, until now, a computer was either too blunt an instrument, or insufficiently powerful, to be of much use in mathematics. But at the same time, the field of mathematics grew and deepened so much that today some questions appear to require additional capabilities beyond the human brain.

“There is a growing consensus that human minds are fundamentally not very good at mathematics, and must be trained,” says Bailey. “Given this fact, the computer can be seen as a perfect complement to humans—we can intuit but not reliably calculate or manipulate; computers are not yet very good at intuition, but are great at calculations and manipulations.”

Although mathematics is said to be a “deductive science,” mathematicians have always used exploration, whether through calculations or pictures, to test ideas and gain intuition, in much the same way that researchers in inductive sciences carry out experiments. Today, this inductive aspect of mathematics has grown through the use of computers, which have vastly increased the amount and type of exploration that can be done. Computers are of course used to ease the burden of lengthy calculations, but they are also used for visualizing mathematical objects, discovering new relationships between such objects, and testing (and especially falsifying) conjectures. A mathematician might also use a computer to explore a result to see whether it is worthwhile to attempt a proof. If it is, then sometimes the computer can give hints about how the proof might proceed. Bailey and Borwein use the term “experimental mathematics” to describe these kinds of uses of the computer in mathematics.